A functional analytic approach for a singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain

Abstract

We consider a sufficiently regular bounded open connected subset $\Omega$ of $\mathbb{R}^n$ such that $0 \in \Omega$ and such that $\mathbb{R}^n \setminus \cl\Omega$ is connected. Then we choose a point $w \in ]0,1[^n$. If $\epsilon$ is a small positive real number, then we define the periodically perforated domain $T(\epsilon) \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\cl(w+\epsilon \Omega +z)$. For each small positive $\epsilon$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(\epsilon)$. More precisely, we consider a Dirichlet condition on the boundary of the set $w+\epsilon \Omega$, and we denote the unique periodic solution of this problem by $u[\epsilon]$. Then we show that (suitable restrictions of) $u[\epsilon]$ can be continued real analytically in the parameter $\epsilon$ around $\epsilon=0$.